29. Doubles of free groups

Agol-Groves-Manning’s Theorem predicts that, for every word-hyperbolic group we can easily construct, every quasiconvex subgroup is separable (otherwise, we would find a non-residually finite hyperbolic group!).

In this section, we use graphs of groups to build new hyperbolic groups:

Combination Theorem (Bestvina & Feighn): If is a quasiconvex malnormal subgroup of hyperbolic groups , then is hyperbolic.

Recall: is called a malnormal subgroup of if it satisfies: if , then .

Example: Let be free, not a proper power. By Lemma 11, is malnormal, so is hyperbolic. As a special case, if is closed surface of even genus , considered as the connected sum of two copies of the closed surface of genus , then by Seifert-van Kampen Theorem, for some .

Question: (a) Which subgroups of are quasiconvex? (b) Which subgroups of are separable?

We will start by trying to answer (b). The following is an outline of the argument: Let be a finite graph so that , let be two copies of . Realize as maps , where . Let be the graph of spaces with vertex spaces , edge space , and edge maps . Then clearly, , and finitely generated subgroups are in correspondence with covering spaces . We can then use similar technique to sections 27 and 28.

Let us now make a few remarks about elevations of loops. Let be a loop in some space , i.e., and . Consider an elevation of :

The conjugacy classes of subgroups of are naturally in bijection with . The degree of the elevation is equal to the degree of the covering map .

Definition: Suppose is a covering map and is an intermediate covering space, i.e., factors through , and we have a diagram

If and are elevations of and the diagram commutes, then we say that descends to .

Let be a finite graph, a finitely generated subgroup and a loop. Let be a covering space corresponding to .

Lemma 29: Consider a finite collection of elevations of to , each of infinite degree. Let be compact. Then for all sufficiently large , there exists an intermediate, finite-sheeted covering space satisfying: (a) embeds in ; (b) every descends to an elevation of degree exactly ; (c) these are pairwise distinct.